EPJ Web of Conferences 92, 020 7 5 (2015)
DOI: 10.1051/epjconf/ 201 5 9 2 020 7 5
C Owned by the authors, published by EDP Sciences, 2015
Application of Nyström method to a Fredholm integral equation describing
induction heating
Josef Rak1,a
1
University of Pardubice, Faculty of Electrical Engineering and Informatics, Studentská 95 532 10 Pardubice
Results were obtained by author’s doctoral study in Charles University in Prague, Faculty of Mathematics and Physics,
Department of Numerical Mathematics
Abstract. An induction heating problem can be described by a Fredholm Integral Equation of the second kind.
The equation is used to compute the eddy current of density. One method for solving such an equation is the
Nyström method. It is based on the approximation of the integral in an equation by the numerical integration rule.
This paper shows application of the Nyström method to an induction heating problem. Results of the Nyström
method are compared with an alternative method.
Key words. integral equation of the second kind, induction heating, collocation method, Nyström method
and continuous function. Since Jeddy is a phasor Jeddy =
(Jeddy,x1 , Jeddy,x2 , Jeddy,x3 ) we can rewrite (1) into three complex integral equations for Jeddy,x1 , Jeddy,x2 and Jeddy,x3 . For
the component Jeddy,x1 we have
Jeddy,x1 (t)
ιJeddy,x1 (x) − κ(x)
dt1 dt2 dt3 = Iext F(x) (3)
r(x, t)
1 Introduction
Ω1
where
F(x) = κ(x)
Ω2
Figure 1. Heated body and coil.
A bounded metal body Ω1 with a Lipschitz-continuous
boundary is heated by an external electromagnetic field
produced by inductor Ω2 (see figure 1). The inductor is
formed by a conductor of general shape and position that
carries harmonic current Iext . The main task is to compute
the eddy current of density Jeddy . In article [8] was derived
following equation
Jeddy (t)
dl(s)
dt1 dt2 dt3 = κ(x)Iext
ιJeddy (x) − κ(x)
r(x, t)
r(x, s)
Ω1
Ω2
(1)
Ω1
Formulas for the remaining components Jeddy,x2 and Jeddy,x3
can be obtained by mere interchanging of indices. The specific Joule losses which are needed to compute the temperature distribution are given by
ω J (x) =
κ(x) =
a
Corresponding author: [email protected]
(4)
where e x1 is unite vector e x1 = (1, 0, 0). With the notation
JR (x) = ReJeddy,x1 (x), JI (x) = ImJeddy,x1 (x), IR = Re(Iext )
and II = Im(Iext ) the equation (3) can be split into equivalent system of two real equations
JI (t)
dt = II F(x),
JR (x) − κ(x)
r(x, t)
Ω1
JR (t)
dt = IR F(x).
(5)
−JI (x) − κ(x)
r(x, t)
where
ωγ(T (x))μ0
,
(2)
4π
r(x, t) is the Euclidean distance, x = (x1 , x2 , x3 ) and t =
(t1 , t2 , t3 ) are points in the metal body, s = (s1 , s2 , s3 ) is
a point at the inductor, ω is angular frequency, γ is electrical conductivity, μ0 is permeability of vacuum and ι is
the complex unit. For each bounded and continuous temperature distribution T , κ(x) is a real, positive, bounded
dl(s).e x1
r(x, s)
where
1
Je (x)Je (x)
γ
(6)
Je (x) =
[ReJeddy,x1 (x)]2 + [ReJeddy,x2 (x)]2 + [ReJeddy,x3 (x)]2 +
+ι [ImJeddy,x1 (x)]2 + [ImJeddy,x2 (x)]2 + [ImJeddy,x3 (x)]2
and Je (x) is complex conjugate to Je (x).
!
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20159202075
EPJ Web of Conferences
it holds Ky(x)
Then for the operator K
= M1 K1 Ny(x).
Using sphere coordinates and lemma 1 with
k(x, t) = r(x, t)−1 and kn (x, t) = rn (x, t)−1 where
2 Existence and uniqueness the of
solution of the induction heating model
Equation (3) belongs to the family of Fredholm integral
equations of the second kind with the diagonal singular
kernel function having the following form
λy(x) − k(x, t)y(t)dt = f (x), x ∈ D, λ 0
(7)
D
where D ⊂ Rm (m ≥ 1) is assumed to be a closed, bounded
and connected set and the kernel function k(x, t) is singular
when x = t. A set is connected if it is not a subset of a
disjoint union of two open sets. Equation (3) describing
induction heating problem is of the form (7) with k(x, t) =
−ικ(x)
r(x,t) , f (x) = −ιIext F(x), y(x) = Jeddy,x1 (x), D = Ω1 and
λ = 1. The operator form of (7) is
(λI − K)y = f
where integral operator K is defined by
Ky(x) =
k(x, t)y(t)dt.
(8)
(9)
D
For showing existence and uniqueness of the solution the
well known Fredholm alternative theorem is used.
Theorem 1 (Fredholm alternative) Let X be a Banach
space, let K : X → X be compact linear operator and
let λ 0. Then either the equation (λI − K)y = 0 has a
non-trivial solution, or the equation (λI − K)y = f has a
solution for all f . In such case the operator (λI − K) has
a bounded inverse.
deBy the last theorem we need to show that operator K
fined as
y(t)
Ky(x)
= −ικ(x)
dt
(10)
r(x, t)
Ω1
is a compact operator on space C(D) and 1 is not its eigenvalue. The proof of compactness can be done using following lemma. It is a well known result from functional
analysis and can be found for example in [1].
Lemma 1 Let D ⊂ Rn be closed and bounded set. Let the
integral operator K be defined by (9). Finally assume that
we can define a sequence of continuous functions kn (x, t)
such that it holds
max |k(x, t) − kn (x, t)|dt → 0, as n → ∞.
(11)
x∈D
D
rn (x, t) =
( f, g)1/κ =
Ω1
(15)
f (t)g(t)
dt.
κ(t)
(16)
can be shown to satisfy
Operator K
1 , y2 )1/κ = −(y1 , Ky
2 )1/κ , for all y1 , y2 ∈ C(Ω1 ). (17)
(Ky
Such an operator is called antisymmetric with respect to
the 1/κ-weighted inner product. The real part of its eigenvalues is equal to 0, so 1 is not an eigenvalue of operator
and by Fredholm alternative is (3) uniquely solvable.
K
3 Nyström method
The Nyström method is based on the approximation of the
integral by the numerical integration rule defined as
v(x)dx ≈
n
ω j v(x j )
(18)
j=1
D
where x j ∈ D are called the node points and ω j weights.
Since the kernel function k(x, t) was assumed to be singular
when x = t, the integration rule cannot be applied directly
to (7). The kernel function k(x, t) is approximated by the
bounded kernel function kn (x, t), n = 1, 2, ..., which coincides with k(x, t) outside a certain neighborhood of x = t.
Then equation (7) is rewritten into the form as in [6].
⎤
⎡
⎢⎢⎢
⎥⎥⎥
⎥⎥⎥
⎢⎢⎢
k(x,
t)dt
k(x, t) y(t) − y(x) dt = f (x).
y(x)
−
λ
−
⎥⎥⎦
⎢⎢⎣
D
D
(19)
By changing kn and k in the second integral on the left hand
side of (19) and using numerical integration rule we obtain
⎤
⎡
⎥⎥⎥
⎢⎢⎢
⎥⎥⎥
⎢⎢⎢
k(x,
t)dt
λ
−
yn (x)−
⎥⎥⎦ ⎢⎢⎣
The last lemma cannot be used directly. Let us define operators K1 , M1 and N from space C(Ω1 ) to space C(Ω1 )
as
y(t)
dt
(12)
K1 y(x) =
r(x, t)
D
−
n
ω j kn (x, x j )[
yn (x j ) − yn (x)] = f (x).
(20)
j=1
Ω1
(13)
(14)
1
n
it is simple to prove that K1 is compact operator. Operators
M and N can be shown to be continuous linear operators.
is compact operator. For showing that 1 is not an
Hence K
1/κ-weighted inner product can be used
eigenvalue of K,
and is defined as
Then K is compact linear operator on C(D).
M1 y(x) = κ(x)y(x)
Ny(x) = Im(y(x)) − ιRe(y(x)).
rn (x, t) = r(x, t), when r(x, t) ≥
rn (x, t) = 1n , when r(x, t) < 1n
Now let us run x through the node points and we get a
system of linear equations for yn (xi )
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⎡
⎤
⎢⎢⎢
⎥⎥⎥
n
⎢⎢⎢
⎥⎥⎥
⎢⎢⎢λ +
ω j kn (xi , x j ) − k(xi , t)dt⎥⎥⎥⎥ y (x )−
⎢⎢⎢
⎥⎥⎦ n i
⎣
j=1
D
Also assume that there exists cD < ∞ such that
g(r(x, t))dt ≤ cD ,
max
x∈D
ji
−
n
ω j kn (xi , x j )
yn (x j ) = f (xi ).
(21)
lim max
ν→0 x∈D
The numerical solution is obtained by the interpolation formula
n
f (x) +
λ+
n
j=1
j=1
ω j kn (x, x j )
yn (x j )
ω j kn (x, x j ) −
.
(22)
k(x, t)dt
The integral can be calculated analytically or by some special numerical integration rule.
The problem is to define the bounded approximation
kn (x, t). To do it, we need more assumptions to the original
kernel function k(x, t). Let r(x, t) be the Euclidean distance
of points x, t ∈ D. Let
RD = max r(x, t).
x,t∈D
Assume that there exists function h ∈ C(D × D) and the
positive non-increasing function g ∈ C(0, ∞) satisfying
lim g(t) = ∞,
t→0+
(23)
such that the kernel function is of the form
k(x, t) = g(r(x, t))h(x, t).
n→∞
(24)
where
gμn (u) =
g(u), if u ≥ μn
g(μn ), if u < μn .
(33)
The last conditions seem to be very restrictive, however
with some effort it can be proven that it is satisfied for several compound numerical integration rules (for example if
D is a rectangle or cuboid then it can be proved to be satisfied for the compound mid-rectanglular or the compound
mid-cuboid rule). The convergence is done by operator calculus. Specifically the by following theorem. It is a modified version of theorem 4.1.1 from [3].
Theorem 2 Let X be a Banach space, let operators S, T
be bounded on X. For given λ 0 let us assume that λI −
T is a bijection on X (which means (λI − T )−1 exists, is
bounded and R(λI − T ) = X).
If
|λ|
(T − S)S <
(34)
(λI − T )−1 then (λI − S)−1 : R(λI − S) → X exists, is bounded and
j=1,...,n
(26)
(27)
(35)
(36)
If S is compact operator then R(λI − S) = X.
First equation (20) needs to be rewritten into operator form.
With the definition
Kn y(x) =
n
ω j kn (x, x j )y(x j )
(37)
j=1
n y(x) =
K
n
ω j kn (x, x j )[y(x j ) − y(x)] +
j=1
+
k(x, t)y(x)dt.
(38)
D
(20) is equivalent to
n )
yn = f.
(λI − K
(29)
where 0 < ρ < ∞ and that there exists μ < ∞ such that
g(μn )ωn ≤ μ for all n.
1 + (λI − T )−1 S
.
|λ| − (λI − T )−1 (T − S)S
y − zX ≤ (λI − S)−1 T y − SyX .
(28)
For the sequence μn assume that
m
μm
n ≥ ρ ωn ,
(λI − S)−1 ≤
Let f ∈ R(λI − S). Let y be solution of (λI − T )y = f and
let z be solution of (λI − S)z = f . Then it holds
Since g ∈ C(0, ∞) was a positive non-increasing function,
gμn ∈ C[0, ∞), is also a positive and non-increasing function for all n. Note that approximation (26) is a bounded
approximation of the original k(x, t).
For the proof of convergence are needed further assumptions to the kernel function and numerical integration
rule. Assume that the numerical integration rule is convergent for all continuous functions and all its weights are
positive. Let us define
ωn = max ω j .
(32)
{t,r(x,t)≤ξ}
(25)
Then kernel function can be approximated by function
kn (x, t) = gμn (r(x, t))h(x, t),
g(r(x, t))dt = 0.
Finally assume that there exists constant c < ∞ such that
for all positive, non-increasing function z ∈ C[0, ∞) and
x ∈ D holds following inequality
ω j z(r(x, x j )) ≤ c z(0)ωn +
z(r(x, t))dt .
Now let us define bounded kernel approximation kn (x, t).
Assume that μn is a decreasing positive sequence such that
lim μn = 0.
{t,r(x,t)<ν}
j,r(x,x j )≤ξ
D
(31)
and
j=1
ji
yn (x) =
{t,r(x,t)<RD }
(30)
(39)
We need to verify condition (34) for T = K and S =
n . It can be done with the theory of collectively compact
K
operators described by Anselone in [5].
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Definition 1 Let X be a Banach space. Let {Kn , n ≥ 1} be
a family of linear operators on X into X. Assume that the
set
{Kn y, n ≥ 1, y ≤ 1}
has compact closure in X. Then {Kn , n ≥ 1} is called collectively compact family of operators on X.
The most important properties of collectively compact operators related to integral equations of the second kind are
summarized in following theorem
Theorem 3 Let X be a Banach space and let K be a linear
operator defined on X into X. Assume that {Kn , n ≥ 1}
is collectively compact family of operators on X. Finally
assume that
Kn y → Ky as n → ∞ for all y ∈ X.
(40)
Then K is compact operator, for any compact operator
M : X → X it holds
(K − Kn )M → 0 as n → ∞
(41)
(K − Kn )Kn → 0 as n → ∞.
(42)
and
Assume that the kernel function is of the form (24),
where g satisfies (23). Also assume that the numerical integration rule converges for all continuous functions and
has positive weights. Finally assume that (29) - (33) hold.
Then the sequence of operators Kn is collectively compact
approximation of K. Since operator K̃n can be written as
n +
From (44) and proposition 1 we have that (λI − K
−1
Kn ) exists as a bijection from C(D) to C(D) for all sufficiently large n. Since the left hand side of (46) is invertible
then the right hand side is also invertible. So the operan + Kn )−1 ] must be one-to-one, othertor I − Kn (λI − K
wise the right hand side of (46) would not be invertible.
n + Kn )−1 is a compact operator it follows
Since Kn (λI − K
from the Fredholm alternative theorem 1 that the operator
n + Kn )−1 ]−1 exists from C(D) into C(D).
[I − Kn (λI − K
From this and (46) we get
n )−1 =
(λI − K
n + Kn )−1 −1 .
n + Kn )−1 I − Kn (λI − K
(λI − K
Both operators on the right hand side of (48) exist as operators from C(D) to C(D). Hence
n ) =
R(λI − K
n + Kn )−1 (λI − K
n + Kn ) = C(D)
= R I − Kn (λI − K
n )−1 exists as an operator from C(D)
and operator (λI − K
to C(D) and we get the existence of a numerical solution
and convergence of the Nyström method.
3.1 Application of Nyström method
For simplicity let us assume that the metal body Ω1 is
cuboid. Let us use the compound mid-cuboid rule. Let xi
be the node points defined as center of sub-cuboides. For
the weight let
n = Kn − (Kn u − Ku)I
K
ωj = ω =
where u(x) = 1 and I is an identical operator, there is no
problem to show that from (42) follows that
n )K
n → 0
(K − K
n → 0.
Kn − K
(43)
(44)
n is not compact. From (36) in theorem
However operator K
n
2 we get for T = K and S = K
n ) Ky − K
n y∞ .
y − yn ∞ ≤ (λI − K
−1
n = I − Kn (λI − K
n + Kn )−1 (λI − K
n + Kn )
λI − K
(46)
and well known proposition from functional analysis.
Proposition 1 (Inverse Theorem proposition) Let X be
a Banach space and let T : X → X be continuous linear
operator. Assume that
1
T < 1.
|λ|
|Ω1 |
, j = 1, ..., n.
n
Let rn (x, t) be approximation of r(x, t) which coincide
outside a certain neighborhood of x = t. The exact construction is immaterial. From (21) we get a system of linear
equations for Jn (xi )
⎤
⎡
⎥⎥⎥
⎢⎢⎢
n
⎢⎢⎢
ω
dt1 dt2 dt3 ⎥⎥⎥⎥ ⎢⎢⎢1 − ικ(xi )
⎥⎥ Jn (xi )+
+ +ικ(xi )
⎢⎢⎢⎣
r (x , x j )
r(xi , t) ⎥⎥⎥⎦
j=1 n i
Ω
1
ji
(45)
But the existence of yn is guaranteed only for f ∈ R(λI −
n ). However we have the following identity
K
(48)
+ικ(xi )
n
ω Jn (x j )
= −ιIext F(xi ), i = 1, ..., n.
r (x , x j )
j=1 n i
(49)
ji
Jn (xi ) is complex number. Let us define for each i = 1, ..., n
Ji(R) = Re Jn (xi ) and Ji(I) = Im Jn (xi ).
Then (49) is equivalent the system
⎤
⎡
⎥⎥⎥
⎢⎢⎢ n
⎥⎥⎥
⎢⎢⎢
ω
dt
dt
dt
1
2
3
(R)
⎥⎥⎥ J(I) −
−
Ji + κ(xi ) ⎢⎢⎢⎢
⎥⎥⎥ i
⎢⎢⎣
r
(x
,
x
)
r(x
,
t)
j
i
⎦
j=1 n i
Ω
ji
(47)
Then (λI − T ) has inverse, which means that (λI − T )−1 :
X → X is a bounded linear operator.
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−κ(xi )
n
ω J(I)
j
j=1
ji
rn (xi , x j )
1
= ImIext F(xi ), i = 1, ..., n
(50)
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5 Example
⎤
⎡
⎥⎥⎥
⎢⎢⎢ n
⎢⎢⎢
ω
dt1 dt2 dt3 ⎥⎥⎥⎥ (R)
(I)
⎢
⎥⎥ J −
− Ji + κ(xi ) ⎢⎢⎢
−
⎢⎣⎢
r (x , x j )
r(xi , t) ⎥⎥⎥⎦ i
j=1 n i
Ω
1
ji
−κ(xi )
n
ω J(R)
j
j=1
ji
rn (xi , x j )
= ReIext F(xi ), i = 1, ..., n
(51)
A brass cuboid body with the size 0.15×0.01×0.01 m (see
figure 2) is heated with a stationary inductor starting at the
room temperature 20 ◦ C. The inductor has the form of a
coil which turns around the heated body in the x1 -direction
in 6 loops. The radius of the coil is 0.015 m, exciting current 500 A and frequency 150 kHz. The length of the coil
is 0.15 m.
Following Nyström interpolation formula (22) we get the
numerical solution for eddy currents Jn
Jn (x) =
−ιIext F(x) − ικ(x)
1 − ικ(x)
where
n
j=1
ω
rn (x,x j )
n
j=1
ω
rn (x,x j ) Jn (x j )
+ ικ(x)
Ω1
,
(52)
1
r(x,t) dt
Jn (xi ) = Ji(R) + ι Ji(I) .
Since since Ω1 is cuboid the integral in (51) and (52) can
be computed analytically.
4 Collocation method
To compare the results of Nyström method we will use
the piecewise constant collocation method described in [7].
Let us cover Ω1 by a regular mesh of n sub-cuboides
Ω1,1 , .., Ω1,n and let the approximation points be the center
points of sub-cuboides. Applying the piecewise constant
collocation we get a numerical solution
Jn (x) =
n
χi (x)Ji
Figure 2. Heating of a brass body - 6 loops
The cuboid is partitioned by 75 elements in x1 direction, 5 elements in x2 and x3 direction. Figure 3 shows the
specific Joule losses distribution calculated by the piecewise constant collocation and the Nyström method with
the compound mid-cuboid integration rule on the x1 axes
with x2 = −0.004 and x3 = −0.004. The same situation
with x3 = 0 is shown in figure 4.
(53)
i=1
where χi = χΩ1,i is a characteristic function of Ω1,i and Ji
is the solution of a system of linear equations
Ji + ι
n
κ(xi )J j
j=1
Ω1, j
1
dt1 dt2 dt3 =
r(xi , t)
= −ιIext F(xi ), i = 1, ..., n.
Ji(R)
(54)
Ji(I)
Let us define for each i = 1, ..., n
= ReJi and
=
ImJi . Then the system of linear equations (54) can be rewritten into system of linear equations for Ji(R) and Ji(I) :
Ji(R)
−
n
j=1
−Ji(I) −
n
j=1
κ(xi )J (I)
j
Ω1, j
κ(xi )J (R)
j
Ω1, j
dt1 dt2 dt3
= ImIext F(xi ), i = 1, ..., n
r(xi , t)
dt1 dt2 dt3
= ReIext F(xi ), i = 1, ..., n.
r(xi , t)
(55)
The numerical solution is (53) with Ji = Ji(R) + ιJi(I) . The
singular integral in the system of linear equations can be
also calculated analytically here.
Figure 3. 6 loops, x2 = −0.004, x3 = −0.004
Computation was made by Matlab. The integration for
computing F(xi ) is done by the Matlab’s method quad. It
uses adaptive the Simpson quadrature. For the computation of non-singular integrals of the matrix of collocation
method simp3D is used. It was developed by W. Padden,
Ch. Macaskill from The University of Sydney in 2008. It
uses compound product Simpson integration rule. Matlab
has the tool triplequad for computing triple integrals, but
it is too slow.
To see the dependence of Joule loses and coil structure
let us make another example with one change. The coil
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Figure 4. 6 loops, x2 = −0.004, x3 = 0
Figure 7. 3 loops, x2 = −0.004, x3 = 0
in the body from the coil is well described by the pictures above. If we compare computing time on operating
system Windows 7 with Inter Core i5, 4GB ram we get
approximately 2 minutes for the Nyström method and approximately 12 minutes for piecewise constant collocation.
This is main and big advantage of the Nyström method. It
reduces time needed for computation.
References
Figure 5. Heating of a brass body - 3 loops
turns around the heated body in the x1 -direction in 3 loops
(see figure 5). Figure 6 shows the specific Joule losses
distribution calculated the by piecewise constant collocation and the Nyström method with the compound midcuboid integration rule on the x1 axes with x2 = −0.004
and x3 = −0.004. The same situation with x3 = 0 is shown
in figure 7.
1. J. Lukes, Zapisky z funkcionalni analyzy (Karolinum,
Praha, 1998)
2. J. Lukes, J. Maly, Mira a Integral (Karolinum, Praha,
1993)
3. K. E. Atkinson, The Numerical Solution of Integral
Equations of the Second Kind (Cambridge University
Press, 1997)
4. K. E. Atkinson, W. Han, Theoretical Numerical Analysis (Springer-Verlag, New York, 2001)
5. P. M. Anselone, Collectively compact operator approximation theory and applications to integral equations
(Prentice-Hall, Inc., Englewood Cliffs, N.J. 1971)
6. L. V. Kantorovich, V. I. Krylov, Approximate Methods
of Higher Analysis, (Interscience New York, 1958)
7. K. Atkinson, I. Graham and I. Sloan, SIAM J. Numer.
Anal. 20, 172-186 (1983)
8. P. Solin, I. Dolezel, M. Skopek, B. Ulrych, SPETO 2001
International Conference, 143-146 (2001)
Figure 6. 3 loops, x2 = −0.004, x3 = −0.004
6 Conclusion
From the pictures we can see that both methods have similar results. Also the dependence on distance of the point
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